Acknowledgment
The financial assistance of the Union Carbide Corp. and the Shell Oil Co. is gratefully acknowledged. literature Cited
(1) Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” pp. 196-8, Wiley, New York, 1960. 2) Carman, P. C., Trans. I m t . Chem. Engrs., London 15,150 (1937). j3) Christopher, R. H., M.S. thesis, University of Rochester, Rochester, N. Y., 1965.
(4) Ergun, S., Chem. Eng. Progr. 48,89 (1952). (5) Fredrickson, A. G., “Principles and Applications of Rheology,” p. 199, Prentice-Hall, Englewood Cliffs, N. J., 1964. (6) Grant, R. P., Ph.D. thesis, University of Rochester, Rochester, N. Y., 1965. ( 7 ) McKelvey, J. M., “Polymer Processing,” p. 69, Wiley, New York, 1962. (8) Middleman, S., Gavis, J., Phys. Fluzds 4, 963 (1961). (9) Sadowski, T. J., Ph.D. thesis, University of Wisconsin, Madison, Wis., 1963. RECEIVED for review January 13, 1965 ACCEPTEDJune 21, 1965
DYNAMICS OF MULTIPHASE FLOW SYSTEMS S
. L. S00,
University of Illinois, Urbana, Ill.
General motion of nonreactive gas-solid systems in potential fields with distribution in size of solid particles and including interactions among particles was formulated through the introduction of “multiphase” generalization, applicable to mean interparticle spacing greater than two diameters. Cases of adiabatic potential flow, laminar boundary layer motion, and electrohydrodynamic slug flow are illustrated and physical significance of results i s discussed. Results demonstrate the feasibility of treating such a flow system rigorously,
THE significance of
multiphase (gas-solid, gas-liquid, or other combinations of a particulate phase and a fluid phase) flow (including fluidization) with regard to chemical and nuclear processes, rocketry, and air pollution control is recognized. A substantial number of theoretical and experimental studies have been reported. However, one basic issue has been avoided: the distribution in the size of particles in the theoretical formulation. Except in very isolated cases of large particles (millimeter size), distribution in the size of particles is unavoidable (25) in most physical systems, including a suspension. Rigorous formulation of gas-solid flow with a distribution in the size of particles is considered in this presentation. The basic theoretical approach includes extending the earlier formulation of the flow system as a continuum (78) with recognized qualifications (79). The rationale is that continuum mechanics of a single-phase fluid amounts to a successful simplification of the classical kinetic theory of a flow system by replacing the coordinates of the phase space with configuration space and transport properties. Any generalized formulation of a gas-solid nonreactive system has as its basis the recognized physical concepts concerning a single particle and a cloud of particles as outlined in Table I. Apart from the obvious definition of a multiphase system (phases of solid, liquid, and gas), it is further recognized here that, from the point of view of “continuum” mechanics of a cloud of particles, particles of different sizes constitute different “phases” ; the nonreactive suspension may consist of one gas and one type of solid material. General Formulation
As the solid particles of different size range constitute different phases from the point of view of continuum mechanics, the previous basic formulations (78) may be generalized. 426
l&EC FUNDAMENTALS
Here we illustrate with the case of nonreactive suspension in which the solid particles are sufficiently numerous; and we define the density of a cloud of particles of size range s as p P ( ’ ) [the density of the solid material is denoted as P ~ ( ~ ) ] ; (1)
p p ( S ) = n(s)m,
where n(’) is the number of solid particles per unit volume, m, is the mass of each particle, and the over-all density of the cloud of solid particles is given by
Basic Concepts Relating to Multiphase Concept for Author Date Single particle 1. Drag of sphere Newton 1686 Stokes 1881 Oseen 1911 2. Apparent mass in oscillatory motion Stokes 1898 3. Relative acceleration in turbulent fluid Lin 1943 Tchen 1947 4. Charge collection Gilbert 1600 5. Thermal electrification so0 1963 Cloud of particles 1. Brownian motion Brown 1882 2. Dispersion and attenuation of Sewell 1910 sound wave 3. Apparent thermodynamic properties Tangren 1949 so0 1961 1962 So0 4. Multiplicity of streamlines Multifurcation of states of tur65.. so0 1962 bu 1ence 1962 Slip velocity at boundary So0 1962 7. Integrals of phase interaction So0 1964 so0
Table 1.
A.
B.
Flow Ref. No. (8) (8) (8) (8)
(70) (28) (30) (22, 23) (6)
(74)
The over-all density of the cloud of Ar solid particles is now given by
(3) where '0 is the volume of the suspension under consideration. The density of the gaseous phase is still (79)
(4) where p is the density of the gas based on its equation of state and p p ( s ) / p p ( s ) is the volume fraction solid of species s. I n such a continuum, these solid particles, in spite of visual discreteness, may be treated as a quasi-continuum and the continuity equation is:
a P p ( S ) / a t4- app(S)upf(s)/aXi = o
+ apu,/ax,
=
o
Fp(Sr)[u P t (r)
LTpi(s))-
-
Jo I,(*)
a/at
+
pp(7)j7p(T
(11)
Km(s)pp(s)dUpz(s)/dtp(S) f aP/bx, =
( b / b ~ i ) ~[ p~ (p ~( ~) )
I
(alax,)pp(S)ep(S) 1 (7)
+ uPrc(s) alaxk
(8)
+ u, alaxn;
(9)
where p is the viscosity of the fluid, FpC(')is the external force acting on particles of sizi: range s, FP('I)is the time constant of the momentum transfer between particle clouds of size range r and size range s. F(,) is the time constant of fluid-particle interaction, a, is the radius, F(,) = (3/8) CD (p/pp)(Au/us) where C, is the drag coefficient at relative speed Au, F(,) = for spherical particles in the Stokes range for single particles or in dilute :suspensions. For substantial particle concentrations, it is readily shown that Ergun's pressure drop equation gives (7) :
CD = ( * / ~ ) ( u F / A u J ( ~ ~=/ ~200(1 ~)
(10)
- E ) E - ~ [ ~ / ~ u ( A u ,f) P ] (7/3
3( a )/ w p ' e 9
-
I j a
+ st + Ck,@)
(2/3) ( a / b ~ ~ ) ~ ~ ( ~ ) e ~ ( ~ ) [ ~ ~ , ( ~ ) ~ ~ ( ~ ) / r (12) n,l
accounting for the fact that particles of different sizes have different streamlines, and:
d/dt =
+d m x )
Thus distribution of particle size in terms of clouds ( r ) is readily treated. When there is multiple scattering of (r) due to its high concentration, the (r) phase would contribute as a viscosity, pP('), given before (78). T h e over-all momentum equation of the system is given by the conservation of momentum of the mixture as:
~ r n ( ' ){
+ [ F p r ( s ) / m , ]+
a/at
pp(S)Fp(ST)
I
where
d/dt,(s) =
'up(') - u p ( r ) l p p ( r )
d m x
( a / b x j ) p ' D j L- ~
[d/dtP(';] [Ut - Upi(a)]
[ b / b ~ , i ~W ~ ( ~) I,( ) -
d O Z ) Z
where m, and m, are the mass of particles (r) and (s), p P ( , ) is the density of the cloud of particles ( r ) based on the volume of the cloud, with
-
[ d / d T p ( s ) ] ( Ui U P f ( S ) ) d 7 p ($1 S)
l/2[p/pP@)]
+
m(
S
= G ( ~ , * / ~( T, )p p ) 1 / 2
T~(')]-''~
a Z
d
da;
pdU,/dt
r
u P p1 + p p ( s ) -w/axl
?p) ( d
(3/4)
(6)
where Ut is the velocity of the gas. T h e equations of motion include that of exchange of momentum between a solid particle and the gas and that of exchange of momentum with the inixture. T h e equation of motion of a single particle in a fluid given by Tchen (28) for spherical solid particles of radius a, is extended to:
dLTpi(s)/dtp(S)- F(S)(Ur-
FP(W =
(5)
where t is the time, x i is the ith component of space coordinate, and Upr(')is the ith component of the velocity of particles of size s. For the gas phase, the continuity equation is:
ap/at
tion. T h e case of interaction of particles in potential motion in a fluid was treated by Hicks, Herman, and Basset (8). Subsequently, the fraction of impaction [ v ( ~ ' )of] a cloud of particles (r) with a body (s) was studied by Langmuir and Blodgett (9), Ranz and Wang (73), and others (3) for both potential and viscous motions; interactions of clouds with or without electrostatic charges was further treated by Kraemer and Johnstone (7). Although these studies were based on stationary targets, generalization to clouds in relative motion is feasible through using the center of mass frame of reference (4). It is readily shown that, under the specification of single scattering,
4
for (1 - E ) = p p / p p , E being the fraction void, AuC = (relative superficial velocity)/e; between 0.92 < E < 0.40. Single particle drag coefficient prevails for E > 0.92, and E ,- 0.40 for packed bed whose pressure drop is given by Chilton and Colburn (76). FP('I) was studied in a preliminary sense by Peskin and So0 ( 7 7 , 72, 24), but can be determined from considering impac-
S
where K,(') is the effectiveness of momentum transfer between the (s) particles and the fluid due to irreversibility in the multi1 for u, < upr (79); phase system, K, = 1 for ut > up?,K, I Siis the ith component of the external force on the fluid such as due to an electric field; SPz(')is that on the (s) particles; Di6 and p' are the deformation tensor and viscosity of the fluid component in the suspension, and [DP(')ljiand pP(') are those of the particles of size s (6, 78).
+ aui/ax, = aupp/ax, + auPp/axr
D,, = (
~
~
(
3
)
)
au,/ax,
~
~
(13) (14)
P is the hydrostatic pressure in the fluid; 8 and 8 , are the dilation of the fluid and the particles:
e
=
au,/ax,
ep(s)= aupk(s)/aXn;
(15 ) (16 )
Since random motion of solid particles is due to the viscous forces exerted by the turbulent field of the fluid and the solid particles have negligible thermal random velocity, the "viscosity" of the particle phase is a quasi-microscopic affair. This, together with other transport properties concerning solid particles, has been discussed (78). T h e energy equations also include the relationships of exchange of energy between a solid particle and the gas and of the exchange of energy with the mixture as a whole; the effect of diffuse radiation (78) is neglected in the present study. T h e energy equation of particles of size s is given by : VOL.
4 NO. 4 N O V E M B E R 1 9 6 5 427
- TP I +
dTp(SI/dtp(s) = G(3)[ T - Tp(s)] +
Gp(sr)[Tp(C
-u=up
@)
I
( b / b ~ t ) ~[bTP(')/bx,] ~(') (17) where T and TP(')are absolute temperatures of the fluid and particles (s), respectively; G(') is the time constant of convective heat transfer between the particle and the fluid, G@)= [ N N u ( S ) ] . ~Zna,/rn; cp is the specific heat of the solid material; [NNucs)] is the Nusselt number of the solid particles in the gas, K is the thermal conductivity and T is the temperature of the gas; G@)= 3~/c,&,a~ for spherical particles in the Stokes law range of relative motion; when applied to dense suspensions, relations given by Ferron and Watson (2) should be followed. Gp(sr)is the time constant of heat transfer between particle clouds of size range r and size range s. For elastic collision of spheres of different sizes, the basic formulation by Herz and Rayleigh (29) gives the deflection, time, and surface area of contact. Heat transfer considerations for uniform temperatures of particles (due to their high thermal conductivity compared to the fluid) lead to a relation between impact and thermal diffusion such that:
;p *
v, v y
GP(W =
C(sr) [ p p ( r ) / 4 p p ( * ) p p ( 7 ) ] [ 4
,
m
/
c
p
(
3
)
4~]/a,ar (18)
with
Figure 1. Boundary layer thickness and distributions of velocity and Concentration of laminar flow over flat plate
and the impaction number N1m(s7),correlating impact and deformation, is
'
1.06
NIm(S') = 5n u p ( , ) - up ( 7 ) 12dpg(()pp(r)X
[
(1
- v,2)(1
-
EsEr
v:)]1/2
(1 -p I*) 2
where and a P ( l ) are the thermal conductivities of the parv,")/n E,, vs is the Poisson ratio and ticle materials, k, = (1 E, is the modulus of elasticity of the material of particle (s), and r* is the ratio of rebound speed to incoming speed. The reciprocity relation is
-
cp(S)pp(3)Gp(S7)=
cpV ) p p ( 7 ) G p(75)
(19)
A further note of interest is the heat transfer a t a wall (s) due to impact of particles (r). T h e heat transfer coefficient is given by
i Figure 2.
45 428
l&EC FUNDAMENTALS
t0.2
Distribution in size of particles given by Equation
E { ( d / W [ P p ( s ) u p P )[ D p ( S ) I t j } E
( 2 / 3 ) ( d / b x j )[
Boundary layer Motion with Distribution in Particle Size
~ ~ : ~ ) u+~ ~ ( ~ )+eS B~ (( 2~1 ) ) ] F~E(~)
S
I n Equation 21, c is thLe specific heat of the gas at constant pressure, K’ is the thermal conductivity of the gas component in the suspension, and K ~ ( ’ ) is the apparent “thermal conductivity’’ of the solid particles (78) in the suspension. T h e include energy dissipated per unit energy fluxes, FE and volume due to the drag on the particles, or:
When applied to incompressible (gas) laminar boundary layer motion of an uncharged suspension of spherical solid particles with distribution in size, the basic equation for steady flow over a flat plate simplifies to, for a light suspension [FP(“)= 0, K,(s) = 0 , P = constant, free stream velocities U = U p = constant, p p o = constant for the free stream, Figure 11,
(bulbx)
p w b x
Adiabatic Potential Flow
Some understanding of the basic nature of the multiphase flow system may be achieved by considering potential motion. For the case of a nonsh,ear flow of a nonreactive, uncharged gas-solid suspension, and neglecting the volume occupied by the solid particles, the right-hand sides of Equations 12, 7, and 21 are equal to zero. These, together with the energy Equation 17 of particles constitute the basic equations of potential motion of a suspension. Since the entropy of a given volume of the suspension is given by:
where R is the gas constant, these four equations can be combined to give the rate of (entropy generation :
[pp(S)up(S)]
+p w b Y
=
=
0
(26)
(ww
(a/+)
+ up(S)bup(S)/by= F(S) [ U P ) -
(27)
UP(S)]
(28)
where u , u p : and x are velocities and coordinate in the x-direction along the plate; u , u p , and y are those normal to the plate. The boundary conditions are: u(x,o) =
u(x,o)
up(S)(x,a ) = u,(’)(x,o) =
=
0,
u, u ( 0 , y ) = u u, up(S)(x,0)= 0
=
U(X,,)
u, u,(S)(o,y) = U - FX = Uplo(‘)
(29)
(the latter from integrating Equation 28 aty = 0 )
pp(S)(x,.,)
=
pp0(S)
pp0(’)U = Lim y+m Ppo =
c
’$
pP(s)up(s)dy
-
Y
0
PPo(s)>P p =
E
c
Pp(s)
S
1
I
1
(30)
In spite of the expected complexity of this problem, the solution is simplified by the above specification of a light suspension; thus the fluid velocity is unaffected by the presence of particles (Equation 2 7 ) and the particles are noninteracting. Therefore the earlier solution based on the momentum integral method (79) for particles of one size can be applied here with additional specification of size s. Thus, at near the leading edge, the density at the wall is now: pplo(S)
% pp0(S){ 1
The slip velocity is, for x
UpzO(S) z
+ [F(S)x/2U]2)
(31)
< U/F, u{1
-
[F(S)x/U]}
(32)
and for fluid boundary layer thickness, 6 (75), 6 =
(23)
(25)
= 0
+ @/by)
( b l d x ) [pp(S)up‘”’1
Up(S)bUp(S)/bX
as well as due to other energy sources. The above equations account for general motion, field forces, heat transfer, and size distribution. For nonspherical particles, time constants F’s and G’s can be determined experimentally. Logical extension can be made to cases involving mass transfer, chemical reaction, etc. Since particle-particle interactions are accounted for, but not the internal stresses of particles, these relations are applicable to concentrations up to the dilute phase fluidized bed, say, down to 90% fracti.on void, but not to dense beds. This calls for a mean interparticle spacing of 2 to 3 particle diameters as a lower limit; for interparticle spacing above 10 diameters, Fp’s and G,’s can be neglected. Extending the generalization of the “multiphase” concept, it is readily seen that particles of different shapes and densities constitute different phases also in the sense of continuum mechanics.
+ (bu/by)
didpx/u
(33)
the particle boundary layer thickness 6 p ( s )is
6,(’) E (1/5) [ F ( ’ ) x / U ] / (1
-
[F(s)x/2U]}
(34)
The above relations suggest that the distribution of sizes of particles at the wall will be very different from that in the free stream. This can be determined as follows: For given size distribution of particle radius a, such as in Figure 2 , in the free stream the numbers of particles in a given volume ‘u are N ( s ) and N ,
d [ S ( S ) / A i ] d a= A f o ( a ) I t is seen that for a dissipative system such as a gas-solid suspension, S > 0. On the basis of the above relations, the speed of sound and apparent therm0dynami.c properties of a suspension of given size distribution can be computed as before (77, 79). For continuous distribution in particle size, of course, the above summations revert to integrals.
where A is a normalizing factor and fa(.) function of radius a , or
(35) is the distribution
where a, is the maximum radius of particles in the distribution. The density of the particle cloud in the free stream is thus: VOL. 4
NO. 4
NOVEMBER 1 9 6 5
429
T 5
5
I
F(Ob2u= 0 (also the free stream)
4-
z3 = pp;
+ ( N / U ) rn,[F(0)x/2U]2Aa,
am
a-lf,(a)da
(38)
where m, is the mass of a particle of characteristic or mean radius a, and
F(O) = 9p/2ppaO2
(39)
T h e corresponding distribution in size at the wall is
d[N,cS)/N,]/da
A,f,(a){ 1
=
+
(ao/a)4[F(0)x/2U]2}( 4 0 )
namely, more small particles at the wall than in the free stream. [F(O)x/U]is the gas-particle momentum number, based on dimension x given earlier (79) and by Heme (5) as "particle parameter." N , is the ratio of inertia force of solid to viscous force of the gas. As an example, we take a distribution in particle size in the free stream as given by:
io(.) = A a exp[-(a - a,)2/(Aa)2]
(41)
where a, is a characteristic or nearly mean particle radius and Au is the range of size variation as indicated in Figure 2. For small ./a,, or sufficiently narrow size distribution such that
Aula, -< 0[10-']
(42)
the integral
lm
f,(u)da 2 m;J
(43)
fo(a)da
and
A
=
l/a,(Aa)
dg
(44)
or, the distribution is given by (Figure 2) :
d(NS)/N)/d(a/a,) =
r-lI2(a/Aa)
exp[- ( a
- ao)z/(Aa)2]
(45)
which gives, from Equation 37, p p 0 = (4*/3)(i\'/V)
PPao3[1- 9(Aa/aO)' 4-( 3 / 4 ) ( A a / d 4 I ( 4 6 )
which shows the deviation in particle density if a, is used without considering distribution in size. The size distribution at the wall is now d ( ~ , ( s ) / ~ , ) / d a = A,{ 1
2t
'L
00
I
I
1.5
2
Electrohydrodynamic Flow into a Channel
To illustrate the application of the basic relations to an electrohydrodynamic flow system of charged solid particles into a grounded channel with low particle concentration (below, say 0.25 kg. per cu. meter), we consider the following problem, where the above equations are simplified from consideration of: two-dimensional motion in an electric field (i = 1, 2); fluid motion not strongly affected by particle motion; and one particle size (s = 1). We consider the case of slug flow of a cloud of charged solid particles with uniform longitudinal ( x direction) velocity L',, = u, of the fluid into a two-dimensional channel of width 26 with grounded conducting walls as shown in Figure 4 . Viscous force encountered by particles drifting toward the walls (at velocity u in the y direction) is taken into account. In this case, the electrostatic forces acting on the particle cloud are entirely due to image charge of the conducting wall and the space charge of the cloud. At the inlet into the channel, the density of solid particles is The collision of particles with the wall is taken uniform (p,,). to be inelastic for the present solution. Taking the range where u p is nearly constant-that is, before the hydrodynamic boundary layer has developed significantly (hence the y
i I
- ~ o ) 2 / ( A a ) z l (47)
giving a greater number of small particles at the wall (Figure 3 ) than in the free stream; A, is the normalizing constant at the wall. The density of particles at the wall is now given by: ppw/ppo =
1
+ [F(')x/2UI2[1- 9(Aa/a,)2 + (3/4)(Aa/a,)4]-1 (48)
showing that the ratio p p m / p p is greater than estimation by taking a, as the only size present, and that solution based on mean radius a, may introduce an error of 10% for spread Aula, N 0.1. 430
I&EC FUNDAMENTALS
25
Figure 3. Shift in size distribution a t wall a t various mean gas-solid momentum numbers
+ ( a 0 / a ) 4 [ ~ ( 0 ) x / 2 ~a] 2 ] exp[-(a
--I
4
-c
\\\
\\\\\\\\\\\\\\\\\\\\\
Figure 4. Flow of uniformly charged cloud of solid particles into channel Walls are grounded conductors
component of fluid velocity is zero)-the in the x direction gives, for x ‘ = x/u,,
( q / m ) E , = bu,Jbx’
equation of motion
-bE,/w
+V ~ ( ~ U ~ / ~ Y Y )
= 0
(49)
where ( q / m ) is the charge to mass ratio of the cloud and E, is the electric field in the x direction. T h e continuity equation now takes the form
(bPp/W
+
( b P P V P l ~ Y )=
0
+ vP(~uplby)
=
(q/m)E,
+F(-d
(51)
b E v P y = (q/m)pp/e
(52)
where e is the permittivity. Equations 50, 51, and 52 can be solved simultaneously to determine the variation of particle concentration p p due to collection at the walls, and variation in u p and Eu due to space charge. Solution is accomplished by introducing a stream function I) such that
-b$/by,
p p u p = b$/bx’
(53) which satisfies Equation 50; and, replacing the independent variables (x’,y) by ( X I , $ ) , we reduce Equation 51 to
bvp/b,c’ = ( q / m ) E ,
(55)
(@/dx’b$)(upeFz ’) =
- (q/m)Ze-leFx’
(56)
which has the particular solution : up = - ( q / m ) % - l (1
- e-FZ‘)/F
(57)
Substitution of Equation 57 with Equation 53 into Equation 50 with
where E u is the electric field in they direction and F = 6irap/m, the time constant of viscous drag based on Stokes’ law; a is the radius of the particle and m is its mass. Since E, = 0, the electric field is given by the Poisson equation in the form:
pp =
= (q/m)/e
Elimination of E v between Equations 54 and 5 5 gives
(50)
where p p varies because o f drift of solid particles. T h e equation of motion in they direction is @uplax’)
and Equation 52 to
- Fv,
P* = P P / P P O
x * ~=
(i)
[1
+ (uo/Fx)e-FZ’“ol[ q / m ) z ( ~ p o b 2 / e uI(u,/Fb) o2) (x/b)
Y* = Y / b (58) we get
Pp*lb(x**)l
(b/bx’)(upeFZ’) = (q/m)E,eFZ’
=
0
(59)
giving the distribution of cloud density from a perturbation solution :
+ x * ~ ) - +~ ~ * ~ [+( lx * ~ ) - +~
p* = (1
]
(60)
the stream function or mass distribution
-6
=
1
PPdY
= ppob(y*(l
or
+ (b/hY*)(P* J P* dY*)
+
x*2)-1
+ cy*2/3) x [I - (1 +
X*~)-~I
(54)
+
]
(61)
and the drift velocity distribution (up/uo) = (u,/Fb) [l - e-(”b/uo)(z/b)] (q/m)2(p,,b2/euo2)x
Y*
Iy*(i
/ 0.4 $4-pp0 b)
0
( a ) Stream Lines of Cloud of
+
}
(q/m)2(ppob2/eUo2)= (Re1eb-l
/ 0.2
’I
x*2)-4]
-x
(62)
These relations are shown in Figure 5. The amount collected is shown between the streamline marked 1.0 and the wall in Figure 5, a. I t is noted that
-0.0
O o ’ -PO=. c - -
+ x*2)-1 + (y*3/3)[1 - ( I +
P
Solid Particles
Y“
(63)
where (Re),b is the “electric Reynolds number” defined by Stuetzer (26) and is the ratio of inertia force of the solid particle to the electrostatic force, and
(uo/Fb) = (‘trm)b (64) is the “gas-solid momentum number” based on dimension b. Expressing it another way, part of the coefficient of Equation 62 can be rewritten as: lvm(Re)eb-’ = (uo/Fb) ( q / m ) ’ ( ~ p o b ~ / e ~=o ~ )
[ ( P P 0 / 4 4(q/m)Yb4P2/w2) 1 [(8s/9) (a2/b2)( P P l P ) 1 [ ( P / P U O b )
(b)
Distribution of Concentration and ‘Transverse Velocity
Figure 5. Behaviosr of slug flow of charged cloud of solid particles into two-dimensional channel
I
= (,Ve,)2[(8~/9)(a2/b2)(~p/~) 1 (Re1b-l (65) where Ne” is the “electroviscous number,” the ratio of electrostatic (space charge) force to viscous force of the fluid, and Reb is the Reynolds number of the channel; the bracket term represents the ratio of cross sections of a particle and the duct, and the density ratio of solid material and the gas. Scaling and simulation (30) can be carried out primarily in terms of the above products rather than individual dimensionless quantities (for small particles with large F the exponential term in Equation 62 approaches zero for finite values of x ) . VOL.
4
NO. 4
NOVEMBER 1 9 6 5
431
matical solution. A feasible method for calculating a laminar flow system can be extended to a turbulent flow system with minimal logical empiricism as shown in the case of a one-phase fluid (15). An immediate instance here is that the generalization in terms of electroviscous number, when applied to turbulent pipe flow of a turbulent suspension, takes the form:
where R is the pipe diameter and D, is the turbulent diffusivity of the solid particles (in place of kinematic viscosity in laminar flow) ; this was confirmed by experiments (21,25). The above electric Reynolds number and electroviscous numbers are in the form for self field bf a cloud of charged particles. For given characteristic electric field E,, they take the form, for example:
X/b
Figure 6. Figure 4
Dimensionless impact rate a t wall of channel in
(Re)& = euo2/E* ( d m )b
(69)
and The collision rate per unit area of the channel wall is given by the product [ p p v ] , a t y * = 1. Equation 53 gives [ppvpIw = [ W / b ~ ’ l ~ *=- ~( l / 2 ) ~ ~(Re)eb-lixrm ~u~
[I
- e - F z / u o ] ( (1 + x * ’ ) - ~
- (4/3)(1
+ x*2)--5 +
,
}
(66)
I t is also seen that ppvp = 0 at x = 0 (the inlet) and the initial rate of increase of [ppvp], is given by [bPpvp/bx’lz=O = Ppouo (Re)eb-l
(67)
regardless of the value of -Vm. These results are plotted as shown in Figure 6 for (Re),b = lop2, and LTm= and noting that for p p o 0.5 kg. per cu. meter (-200 grainsper cu. foot), b = 0.02 meter, uo = 2 meters per second, q / m = 3 X coulomb per kg., (Re)eb 0.02 for 1-micron particlesof specific gravity of 2.5 in air at room condition; F N l o 5 sec.-l for b = 0.02 meter and uo = 2 meters per second, giving E,], X 105 v i m and ATm= The corresponding -2.5 flow Reynolds number, (Re)b, is 2560; .Vev = 2900 for (8?r/9) ( a 2 / b 2 ) ( p p / p )of 1.53 X Figure 6 shows that with higher flow velocity (turbulent flow), to a limit to be determined, the collection rate per unit area could be higher than proportionate to the low velocity (laminar) flow. Extension of basic relations for the study of turbulent floiv remains to be thoroughly carried out. For a particle cloud of a range of size distribution, the above should be modified according to the basic equations, but the above transformation will not be applicable because of multiplicity of stream lines, and mutually self-consistent field. Numerical solution, however, is feasible. A curve such as those shown in Figure 6 will be obtained for each narrow size range; summation gives the total accumulation. This will give proportionately greater amounts of large particles at near the entrance than in the entering suspension, a fact which is well known (30) but can now be computed rigorously.
-
-
Discussion
The above study demonstrates that the general aspects of hydrodynamics and electrohydrodynamics of a multiphase system can be formulated rigorously and illustrates the physical significance of the basic considerations in terms of simplest nontrivial problems. Although in most applications, turbulent flow of suspension, or particle-induced turbulence in the fluid phase (such as in a fluidized bed) are encountered, laminar flow systems were illustrated because they lend themselves to rigorous mathe432
I&EC FUNDAMENTALS
(2Yev)b =
[(&p/4*em)
( b 3 ~ ’ / ’ ~1’’’)
(70)
Even without electric charges, the hydrodynamic equation gives, for the flow of a suspension over a flat plate, continuous deposition of solid particles as the slip velocity of solid particles becomes zero. This is obviously an oversimplification. Although one might account for the actual situation with a “probability of collection” (30),Brownian motion (6) of solid particles should be logically accounted for in the boundary condition. This is because for particles of, say, 1-micron diameter and specific gravity of 2.5, the mean thermal random velocity is 0.3 cm. per second at 300’ K., but for particles of 0.1-micron diameter, this value goes up to 3 meters per second. These magnitudes of velocity are significant at the boundary of the flow system. At the boundary the density of the cloud of particles is given by:
where D is the diffusivity due to Brownian motion, and D = kT/S?rag
(72)
k is the Boltzmann constant, and U d is the drift velocity at the wall. This amounts to a new “diffusion boundary layer thickness” for micron and submicron particles to be added to the above boundary layer solution. The Brownian motion accounts for less effective collection of these small particles in both mechanical and electrostatic precipitators. The above examples illustrate further that the problems of multiphase flow are, in general, complicated and in most cases we have to be satisfied with numerical solutions. However, nontrivial analytical solutions should be sought after to further our general understanding of multiphase systems; this calls for new mathematical techniques beyond those available for onephase fluid. An important area of study will be the determination of transport properties in connection with multiphase systems, and definitive experiments are needed. A natural extension of the above “multiphase” concept will be treating systems involving mass transfer and chemical reaction. Conclusions
Rigorous formulation of dynamics and electrodynamics of multiphase systems is feasible. The generalized “multiphase” concept is such that even for a gas-solid suspension of one type of solid material, particles of different sizes, shapes, masses,
electric charges, dipoles, and magnetism constitute different “phases” in addition to the gas phase. literature Cited
(1) Ergun, D., Chem. E q . Progr. 48, 89 (1952). (2) Ferron, J. R., Watson, C. C., Ibid., Symp. Ser. 58, 79 (1962). (3) Fuchs, N. A., “The Mechanics of Aerosols,” pp. 159, 181, Macmillan? New Yorlr, 1964. (4) Goldstein, H., “Claasical Mechanics,” p. 58, Addison-Wesley, Reading, Mass., 1950. (5) Herne: H., Intern. J . Air Pollution 3, 26 (1960). (6) Kennard, E. H., “Kinetic Theory of Gases,” p. 280, McGrawHill, New York, 1938 (7) Kraemer, H. J., Johnstone, H . F., Znd. Eng. Chem. 47, 2426 (1955). (8) Lamb, H., “Hydrodynamics,” Dover Publications, New York, 1932. (9) Langmuir, I., Blodgsett, K. B., General Electric Rept. RL-225 (1945); J . Meteor. 5, 175 (1948). (10) Lin, C. C., Quart. J . A@. M a t h . 1, 43 (1943). (11) Peskin, R. L.. “Diffusivity of Small Suspended Particles in Turbulent Fluids,” Heat Transfer and Fluid Mechanics Institute. 1960. (12) Peskin, R. L., “Some Effects of Particle-Particle and ParticleFluid Interaction in Two-Phase Flow Systems,” Ph. D. thesis, Princeton University, Princeton, N. J., 1959. (13) Ranz. IV., IL’ang, J., Znd. Eng. Chem. 44, 1371 (1952). (14) Richardson, E. G., “‘Ultrasonic Physics,” Elsevier, New York, 1952. (151 Schlichting, H., “Eloundary Layer Theory,” McGraw-Hill, h e w York, 1960. (16) Sherwood, T. K., .Pigford, R. L., “Absorption and Extraction.” p. 237, McGraw-Hill, New York, 1952. (17) Soo, S. L., A.I.Ch.LI. J. 7 , 384 (1961).
(18) Soo, S. L., “Boundary Layer Motion of a Gas-Solid Suspension,” Proc. Symposium on Interaction between Fluids and Particles, p. 50, Inst. of Chem. Engrs., London, 1962. (19) Soo, S. L., “Gas-Solid Flow,” Proc. Symposium on Singleand Multi-Component Flow Processes, Rutgers Engineering Centennial, Rutgers University, New Brunswick, N. J., May 1, 1964 (in press). (20) Soo, S. L., IND.ENG.CHEM.FUNDAMENTALS 1, 33 (1962). (21) Ibtd., 3, 75 (1964). (22) Soo, S. L., Phys. Fluids 6 , 145 (1963). (23) Soo, S. L., Dimick, R. C., “Experimental Study of Thermal Electrification of a Gas-Solid System,” Multi-Phase Flow Symposium, p. 43, ASME, 1963. (24) Soo, S. L., Peskin, R. L., “Statistical Distribution of Solid Phase in Two-Phase Turbulent Motion,” Proiect SQUID Tech. Rept. PR-80-R (ONR) (1958). (25) Soo. S. L.. Trezek. G. J.. Dimick. R. C.. Hohnstreiter, G. F.. ‘ I k . ENG.CHEM.FUNDAMENTALS 3, 98 (1964). (26) Stuetzer, 0.M., Phys. Fluids 5, 534 (1962). (27) Tangren, R. F.. Dodge, C. H., Seifert, H. S., J . Appl. Phys. 20, 637 (1949). (28) Tchen. C. M., “Mean Value and Correlation Problems ‘ Connected with the Motion of Small Particles in a Turbulent Fluid,” dissertation, Delft, Martinus Nijhoff, The Hague, 1947. (29) Timoshenko, S., “Theory of Elasticity,” p. 339, McGrawHill, New York, 1934. (30) White, J. A., “Industrial Electrostatic Precipitation,” Addison-\Yesky, Reading, Mass., 1963. RECEIVED for review September 16, 1964 ACCEPTED April 19, 1965 Work sponsored by Project SQUID, supported by the Office of Naval Research, Department of the Navy, under contract Nonr 3623 (S-6), NR-098-038. Reproduction in full or in part is permitted for any use of the United States Government.
CON CENT R A T ION POLAR E A T ION EFFECTS IN A REVERSE OSMOSIS SYSTEM W I L L I A M
N . G I L L
Clarkson College of Technology, Potsdam, N. Y . C H I T l E N AND
DALE W .
ZEH
Department of Chemical Engineering and Metallurgy, Syracuse University, Syracuse, A’.
Y.
An analysis was made to determine the polarization of salt concentration a t the membrane boundary in a continuous reverse osmosis system. The results indicated that earlier studies based on constant transverse water velocity along the membrane surface tend to overestimate this phenomenon and consequently give a conservative evaluation of the economic feasibility of reverse osmosis for desalination purposes. When variable wall velocity is taken into consideration, the boundary condition at the membrane surface for the diffusion equation is nonlinear. A perturbation method is used to solve this nonlinear problem such that results can b e related directly to parameters defined in terms of operating variables.
c
effort has been directed toward the study of desalination of sea water in recent years. Reverse osmosis has been considered as a possible method primarily because of its simplicity. T h e essence of a reverse osmosis ONSIDERABLE
system involves the use of selective membranes which permit the passage of \\ater but not salt through them. In practical design, the process probably will utilize a stack of flat membranes srparated by narrow passages through which brine and water nil1 flow in alternate sections as shown in F i”m r e 1. O n e of the most seriow problems encountered in a continuous reverse osmosis system i:, the build-up of salt concentration a t the \Val1 of the membrane along the axial direction as a result
water
(low pressure)
tttfttttttfttltt brine
-*
(high pressure)
‘ : r?/.!..’/ ?, ,?/ !A : !/+/ /+/M/+/ 4
t + + + + + + & & I
-,,,,.,,,,,,,, )/
t
wsotet
+
+ - f
hdw+p!e/strL)+
, + /
brine
+
,? ! 5 ? /
’
‘
+
(high pressure)
,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,~ t I t O t t t t t t t t t + t + k i t t t ( t 4
+ +
woter ~i~~~~ 1.
(low pressure)
Schematic arrangement
of
osmosis
system for desalination VOL. 4
NO. 4
NOVEMBER 1 9 6 5
433
